By N. Balakrishnan
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Characterization of the optimal filter: The non Markov case, Preprint. B. (1965). Markov Processes, Vols. I, II, Springer-Verlag, Berlin. N. G. (1986). Markov Processes: Characterization and Convergence, John Wiley & Sons, New York. , Kallianpur, G. and Kunita, H. (1972). Stochastic differential equations for the nonlinear filtering problem, Osaka Journal of Mathematics, 9, 19–40. W. (1976). Differential Topology, Springer-Verlag, New York. Horowitz, J. L. (1990). ), pp. 75–122, Birkhaüser, Boston.
MANDAL . 12) Therefore, C1 is satisfied by A0. To see that C2 holds, note that This will imply the existence of a countable set that such and hence C2 follows. That D(A0) is an algebra follows from Mohammed (1984, p. 107). It is also easy to check that D(A0) separates points in C[-r, 0] and contains the constant functions which implies that A0 satisfies C3. 2 we then have that πtX is a solution to the martingale problem corresponding to (A0, η). We now show that it is the unique solution. 10). Then the martingale problem for (A0, η) is well posed.
For 0ՅsՅtՅT, and a C-valued random variable η on (⍀, , P), let and For any Banach space B with norm we denote by L2(⍀, B) the space of all random variables taking values in B such that . For each sample path of a real valued process (t), -rՅtՅT, define The following theorem on the existence and the uniqueness of the solution of an SDDE has been proved by Mohammed (1984). 2(ii) on page 143. 1 Suppose that (⍀, , P), W, are given as above. Suppose are two Borel measurable functions satisfying the following Lipschitz and growth conditions.
Advances on Theoretical and Methodological Aspects of Probability and Statistics by N. Balakrishnan